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 Academic Year: | 2017/8 |
| Owning Department/School: | Department of Physics |
| Credits: | 12 [equivalent to 24 CATS credits] |
| Notional Study Hours: | 240 |
| Level: | Certificate (FHEQ level 4) |
| Period: |
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| Assessment Summary: | CW 20%, EX 60%, OT 20% |
| Assessment Detail: |
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| Supplementary Assessment: |
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| Requisites: | |
| Description:
| Aims: The aim of this unit is to introduce mathematical techniques required by physical science students, both by showing the application of A-level mathematics content to physical problems in a more general and algebraic form and by introducing more advanced topics. Learning Outcomes: After taking this unit the student should be able to: * sketch graphs of standard functions and their inverses; * evaluate the derivative of a function and the partial derivative of a function of two or more variables; * write down the Taylor series approximation to a function; * represent complex numbers in Cartesian, polar and exponential forms, and convert between these forms; * calculate the magnitude of a vector, and the scalar and vector products of two vectors; * solve simple geometrical problems using vectors. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Preliminary calculus (6 hours): Differentiation: differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz' theorem. Integration: integration from first principles; sinusoidal functions; logarithmic integration; using partial fractions; substitution method; by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications. Probability and Distributions (3 hours): Probability; permutations and combinations. Discrete distributions: mean and variance; expectation values; binomial and Poisson distributions. Continuous distributions: expectation values and moments; Gaussian distribution; simple applications, e.g. velocity distributions. Central limit theorem. Series and limits (3 hours): Summation of series: arithmetic, geometric, arithmetico-geometric series; difference method; series involving natural numbers; transformation of series. Convergence of infinite series: absolute and conditional convergence; alternating series test. Operations with series. Power series: convergence; operations with power series. Taylor series: Taylor's theorem; approximation errors; standard Maclaurin series. Evaluation of limits. Complex numbers and hyperbolic functions (4 hours): Manipulation of complex numbers: addition and subtraction; modulus and argument; multiplication; complex conjugate; division. Polar representation of complex numbers, multiplication and division. De Moivre's theorem: trigonometric identities; nth roots of unity; solving polynomial equations. Complex logarithms and complex powers. Applications to differentiation and integration. Hyperbolic functions: definitions; hyperbolic-trigonometric analogies; identities; solving hyperbolic equations; inverses; calculus of hyperbolic functions. Partial differentiation (4 hours): Total differential; total derivative; exact and inexact differentials; useful theorems of partial differentiation; chain rule; change of variables; Taylor's theorem for many-variable functions; stationary values of many-variable functions; thermodynamic relations; differentiation of integrals; least squares fits. Multiple integrals (4 hours): Double and triple integrals. Applications of multiple integrals: areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions. Change of variables in multiple integrals. General properties of Jacobians. Vector algebra (2 hours): Basis vectors and components. Magnitude of a vector. Multiplication of vectors: scalar product; vector product; scalar triple product; vector triple product. Equations of lines, planes and spheres. Using vectors to find distances: point to line; point to plane; line to line; line to plane. Reciprocal vectors. Matrices and vector spaces (7 hours): Vector spaces: basis vectors; inner product. Linear operators. Basic matrix algebra. Functions of matrices. Transpose; complex and Hermitian conjugates; trace; determinant; properties of determinants. Inverse of a matrix; rank of a matrix. Special types of square matrix: diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal. Eigenvectors and eigenvalues of normal, Hermitian and anti-Hermitian, unitary, and general square matrices. Determination of eigenvalues and eigenvectors: degenerate eigenvalues. Change of basis and similarity transformations. Diagonalization of matrices. Quadratic and Hermitian forms: stationary properties of the eigenvectors; quadratic surfaces. Simultaneous linear equations: range; null space; N simultaneous linear equations in N unknowns; singular value decomposition. First-order ordinary differential equations (4 hours): General form of solution. First-degree first-order equations: separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations. Higher-degree first-order equations: equations soluble for p; for x; for y; Clairaut's equation. Normal modes (2 hours): Typical oscillatory systems; symmetry and normal modes. Higher-order ordinary differential equations (5 hours): Linear equations with constant coefficients: complementary function, particular integral, general solution; linear recurrence relations; Laplace transform method. Linear equations with variable coefficients: The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations. General ordinary differential equations: dependent variable absent; independent variable absent; non-linear exact equations; isobaric or homogeneous equations. |
| Programme availability: |
PH10007 is a Designated Essential Unit on the following programmes:Department of Physics
PH10007 is Compulsory on the following programmes:Programmes in Natural Sciences
PH10007 is Optional on the following programmes:Department of Chemistry
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